Find the volumes of the solids in Exercises 65. 66. z = 4 – 4 (x² + y²) z = 1- r z = (x2 + y²)² – 1 z = -V1 67. 68. z = Vx² + y² r = 3 cos 0 r = -3 cos 0

Q: Find the work done by a force F = 6ỉ + 5j– 3k when a particle moves from point P(-3, 4, 5) to point…

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Q: Let G = {(x, y): 1 ≤ x^2 + y^2 ≤ 16,0 ≤ y ≤ x). Sketch G, and find the coordinates of its centroid.

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Q: Compute the flux of F = xi+yj+zk through the curved surface of the cylinder x² + y² = 1 bounded…

A: The given vector field is F→=xi+yj+zk. The goal is to evaluate the flux or surface integral of the…

Q: Integrate h(x, y) = yi-xj over the triangle with vertices (-2, 0), (1, 0), and (0, 1) traversed…

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Q: A fluid motion is given by v=(y+z)i+(z+x)j+(x+y)k show that the motion is irrotational and hence…

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Q: The four corners of a thin homogeneous and square plate with side 2a are folded up so that they…

A: It is given that the side length is 2a.Let the total mass of the system is M.

Q: (a)Show that the vectors a = 2i + 4j8k, b = 3i -j + 3k and c = -5i + 11j - 25k are coplanar.

A: Question No. (a): To show that the vectors are coplanar.Note: Since you have posted multiple…

Q: How can I find the centroid of y2= 4x, y = 2x, using only solving for the equations Mx, My, and A.…

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Q: The volume of the solid bounded by the plane 3x+6y-2z=12 and the coordinate planes is O A. 2-2y+4 1…

A: We need to find the volume of the solid bounded by the plane 3x+6y-2z=12. Now,…

Q: Find the volume of the solid by subtracting two volumes. the solid enclosed by the parabolic…

A: Given that the solid enclosed by the parabolic cylinders and the planes To find the volume of the…

Q: (B) By using the cylindrical coordinates, evaluate fff z √√x² + y² dV where G is the solid enclosed…

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Q: Compute the flux of F = xi+yj+zk through the curved surface of the cylinder x² + y² = 1 bounded…

A: The vector field is F=xi^+yj^+zk^. The curves surface is x2+y2=1 bounded below by the plane x+y+z=2…

Q: 3 dx a) 3- x

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: Find the volume of the solid within the cylinder x² + y? = 9 that is bounded below by the xy-plane…

A: Using cylindrical coordinates the volume of a solid is given by V=∬∫v r dz dr dθ

Q: Find the flux of G(x, y, z) = (xz)i+ (yz)j+(z²)k outward across the part of the sphere x² + y² +z² =…

A: Here, surface, S={(x,y,z): x2+y2+z2=9, and 1≤z≤2}. For z=1, we get x2+y2=8 and for z=2, we get…

Q: 12) find the indefinite inteyral Inv dv 6 v3 Answery a) -L (2invtl /Ac 12 v2 bり- りよemvtiリ+c (2 12v2…

A: The given integral is ln(v)/(6v3).

Q: 4) Evaluate , z dV using rectangular coordinates, where Q is the solid in the first octant bounded…

A: In the first octant,  x≥0y≥0z≥0    Thus x, y and z cannot be less than 0

Q: A) Show that the function u(x; y) = 5x4y - 10 x²y³ + y5 is harmonic and determine the harmonic…

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Q: - Cone o Find the volume of the solid region above the cone (zv3)" (z 2 0) and below the sphere x² +…

A: Using spherical coordinates,

Q: Let V be a region in R3 bounded above by the plane z = 1 and below by the paraboloid z = x² + y².…

A: Given Information:V is the region on  bounded above by the plane z=1 and below by the paraboloid…

Q: Calculate % F-dr where

A: Given,           F=xyi+2zj-xykand C is the intersection of the of the plane y=z+2 withcylinder…

Q: The complex potential of a line vortex of strength k centred on the origin is W=iklnz. Determine the…

A: The objective of this question is to find the complex potential of two line vortices of the same…

Q: Find the orthogonal trajectories of the conicoid (x + y)z = 1 of the conics in which it is cut by…

A: Given:The equation of conicoid is (x+y)z=1The system of the plane is x-y+z=K, where k is…

Q: A quadrivector AH has components (1, 1, 1, 0, 1)' in the system S. Calculate the T components of AH…

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